THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω→ R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|p* |u|p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|p+|u|p* + a(x)), (2) where L≥1, 1pN,p* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.